Okay, you've verified that the coils are good and everything is okay back to the drive transistors. I'm not entirely sure what you mean by the coils showing up in the switch test. Do the slings fire when you close the switches that stick up through the playfield? If they do, then everything is working as it should. High Speed has separate scoring switches for the slingshots, and if you aren't getting any scoring, these could be your problem.
I'm pointing to the scoring switch in this picture, and they are switch 49 and 50 in the switch test. If neither one is registering, check the star rollovers at the top of the playfield, they are on the same column of the switch matrix.